Abstract

A general method for measuring the birefringence of nondichroic, linear retarder media has been presented. The method is based on phase-stepping imaging polarimetry and permits the azimuth angle, phase retardation, and transmission coefficient of a sample to be calculated. The method uses a simple setup, a sample at rest, and permits fast acquisition of data. With the mathematical description applied, various algorithms for different optical configurations can be used and any number of intensity patterns can be generated. Experimental results for photoelastic samples and the results of measurements of the birefringence of optical components and biological samples are also presented.

© 1999 Optical Society of America

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References

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  1. R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
    [CrossRef] [PubMed]
  2. Y. Otani, T. Shimada, T. Yoshizawa, “The local-sampling phase shifting technique for precise two-dimensional birefringence measurement,” Opt. Rev. 1, 103–106 (1994).
    [CrossRef]
  3. A. Asundi, “Photoelasticity and moire,” in Optical Measurement Technique and Applications, P. K. Rastogi, ed. (Artech House, Boston, Mass., 1997), pp. 183–213.
  4. S. J. Haake, E. A. Patterson, “Photoelastic analysis of frozen stressed specimens using spectral contents analysis,” Exp. Mech. 32, 266–272 (1992).
    [CrossRef]
  5. A. D. Nurse, “Full-field automated photoelasticity by use of a three-wavelength approach to phase stepping,” Appl. Opt. 36, 5781–5786 (1997).
    [CrossRef] [PubMed]
  6. C. Quan, P. J. Bryanston-Cross, T. R. Judge, “Photoelasticity stress analysis using carrier fringe and FFT techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
    [CrossRef]
  7. K. Oka, J. Ikeda, Y. Ohtsuka, “Novel polarimetric technique exploring spatiotemporal birefringent response of an anti-ferroelectric liquid crystal cell,” J. Mod. Opt. 40, 1713–1723 (1993).
    [CrossRef]
  8. Z. F. Wang, E. A. Patterson, “Use of phase stepping with demodulation of fuzzy sets for birefringence measurement,” Opt. Lasers Eng. 22, 91–104 (1995).
    [CrossRef]
  9. J. A. Quiroga, A. Gonzalez-Cano, “Phase measuring algorithm for extraction of isochromatics of photoelastic fringe patterns,” Appl. Opt. 36, 8397–8402 (1997).
    [CrossRef]
  10. T. W. Ng, “Derivation of retardation phase in computer-aided photoelasticity by using carrier fringe phase shifting,” Appl. Opt. 36, 8259–8263 (1997).
    [CrossRef]
  11. M. Noguchi, T. Ishikawa, M. Ohno, S. Tachihara, “Measurements of 2-D birefringence distribution,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation, J. Tsujiuchi, ed., Proc. SPIE1720, 367–378 (1992).
    [CrossRef]
  12. P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).
    [CrossRef]
  13. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 26, 350–352 (1984).
  14. K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurements Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 94–140.
  15. J. Jaronski, “Theoretical and experimental study on corneal birefringence,” Ph.D. dissertation (Institute of Physics, Wrocław University of Technology, Wrocław, Poland, 1997).

1997 (3)

1995 (2)

Z. F. Wang, E. A. Patterson, “Use of phase stepping with demodulation of fuzzy sets for birefringence measurement,” Opt. Lasers Eng. 22, 91–104 (1995).
[CrossRef]

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

1994 (1)

Y. Otani, T. Shimada, T. Yoshizawa, “The local-sampling phase shifting technique for precise two-dimensional birefringence measurement,” Opt. Rev. 1, 103–106 (1994).
[CrossRef]

1993 (2)

C. Quan, P. J. Bryanston-Cross, T. R. Judge, “Photoelasticity stress analysis using carrier fringe and FFT techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

K. Oka, J. Ikeda, Y. Ohtsuka, “Novel polarimetric technique exploring spatiotemporal birefringent response of an anti-ferroelectric liquid crystal cell,” J. Mod. Opt. 40, 1713–1723 (1993).
[CrossRef]

1992 (1)

S. J. Haake, E. A. Patterson, “Photoelastic analysis of frozen stressed specimens using spectral contents analysis,” Exp. Mech. 32, 266–272 (1992).
[CrossRef]

1984 (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 26, 350–352 (1984).

Asundi, A.

A. Asundi, “Photoelasticity and moire,” in Optical Measurement Technique and Applications, P. K. Rastogi, ed. (Artech House, Boston, Mass., 1997), pp. 183–213.

Bryanston-Cross, P. J.

C. Quan, P. J. Bryanston-Cross, T. R. Judge, “Photoelasticity stress analysis using carrier fringe and FFT techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

Creath, K.

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurements Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 94–140.

Gdoutos, E. E.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).
[CrossRef]

Gonzalez-Cano, A.

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 26, 350–352 (1984).

Haake, S. J.

S. J. Haake, E. A. Patterson, “Photoelastic analysis of frozen stressed specimens using spectral contents analysis,” Exp. Mech. 32, 266–272 (1992).
[CrossRef]

Ikeda, J.

K. Oka, J. Ikeda, Y. Ohtsuka, “Novel polarimetric technique exploring spatiotemporal birefringent response of an anti-ferroelectric liquid crystal cell,” J. Mod. Opt. 40, 1713–1723 (1993).
[CrossRef]

Ishikawa, T.

M. Noguchi, T. Ishikawa, M. Ohno, S. Tachihara, “Measurements of 2-D birefringence distribution,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation, J. Tsujiuchi, ed., Proc. SPIE1720, 367–378 (1992).
[CrossRef]

Jaronski, J.

J. Jaronski, “Theoretical and experimental study on corneal birefringence,” Ph.D. dissertation (Institute of Physics, Wrocław University of Technology, Wrocław, Poland, 1997).

Judge, T. R.

C. Quan, P. J. Bryanston-Cross, T. R. Judge, “Photoelasticity stress analysis using carrier fringe and FFT techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

Mei, G.

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

Ng, T. W.

Noguchi, M.

M. Noguchi, T. Ishikawa, M. Ohno, S. Tachihara, “Measurements of 2-D birefringence distribution,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation, J. Tsujiuchi, ed., Proc. SPIE1720, 367–378 (1992).
[CrossRef]

Nurse, A. D.

Ohno, M.

M. Noguchi, T. Ishikawa, M. Ohno, S. Tachihara, “Measurements of 2-D birefringence distribution,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation, J. Tsujiuchi, ed., Proc. SPIE1720, 367–378 (1992).
[CrossRef]

Ohtsuka, Y.

K. Oka, J. Ikeda, Y. Ohtsuka, “Novel polarimetric technique exploring spatiotemporal birefringent response of an anti-ferroelectric liquid crystal cell,” J. Mod. Opt. 40, 1713–1723 (1993).
[CrossRef]

Oka, K.

K. Oka, J. Ikeda, Y. Ohtsuka, “Novel polarimetric technique exploring spatiotemporal birefringent response of an anti-ferroelectric liquid crystal cell,” J. Mod. Opt. 40, 1713–1723 (1993).
[CrossRef]

Oldenbourg, R.

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

Otani, Y.

Y. Otani, T. Shimada, T. Yoshizawa, “The local-sampling phase shifting technique for precise two-dimensional birefringence measurement,” Opt. Rev. 1, 103–106 (1994).
[CrossRef]

Patterson, E. A.

Z. F. Wang, E. A. Patterson, “Use of phase stepping with demodulation of fuzzy sets for birefringence measurement,” Opt. Lasers Eng. 22, 91–104 (1995).
[CrossRef]

S. J. Haake, E. A. Patterson, “Photoelastic analysis of frozen stressed specimens using spectral contents analysis,” Exp. Mech. 32, 266–272 (1992).
[CrossRef]

Quan, C.

C. Quan, P. J. Bryanston-Cross, T. R. Judge, “Photoelasticity stress analysis using carrier fringe and FFT techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

Quiroga, J. A.

Shimada, T.

Y. Otani, T. Shimada, T. Yoshizawa, “The local-sampling phase shifting technique for precise two-dimensional birefringence measurement,” Opt. Rev. 1, 103–106 (1994).
[CrossRef]

Tachihara, S.

M. Noguchi, T. Ishikawa, M. Ohno, S. Tachihara, “Measurements of 2-D birefringence distribution,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation, J. Tsujiuchi, ed., Proc. SPIE1720, 367–378 (1992).
[CrossRef]

Theocaris, P. S.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).
[CrossRef]

Wang, Z. F.

Z. F. Wang, E. A. Patterson, “Use of phase stepping with demodulation of fuzzy sets for birefringence measurement,” Opt. Lasers Eng. 22, 91–104 (1995).
[CrossRef]

Yoshizawa, T.

Y. Otani, T. Shimada, T. Yoshizawa, “The local-sampling phase shifting technique for precise two-dimensional birefringence measurement,” Opt. Rev. 1, 103–106 (1994).
[CrossRef]

Appl. Opt. (3)

Exp. Mech. (1)

S. J. Haake, E. A. Patterson, “Photoelastic analysis of frozen stressed specimens using spectral contents analysis,” Exp. Mech. 32, 266–272 (1992).
[CrossRef]

J. Microsc. (1)

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

K. Oka, J. Ikeda, Y. Ohtsuka, “Novel polarimetric technique exploring spatiotemporal birefringent response of an anti-ferroelectric liquid crystal cell,” J. Mod. Opt. 40, 1713–1723 (1993).
[CrossRef]

Opt. Eng. (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 26, 350–352 (1984).

Opt. Lasers Eng. (2)

Z. F. Wang, E. A. Patterson, “Use of phase stepping with demodulation of fuzzy sets for birefringence measurement,” Opt. Lasers Eng. 22, 91–104 (1995).
[CrossRef]

C. Quan, P. J. Bryanston-Cross, T. R. Judge, “Photoelasticity stress analysis using carrier fringe and FFT techniques,” Opt. Lasers Eng. 18, 79–108 (1993).
[CrossRef]

Opt. Rev. (1)

Y. Otani, T. Shimada, T. Yoshizawa, “The local-sampling phase shifting technique for precise two-dimensional birefringence measurement,” Opt. Rev. 1, 103–106 (1994).
[CrossRef]

Other (5)

A. Asundi, “Photoelasticity and moire,” in Optical Measurement Technique and Applications, P. K. Rastogi, ed. (Artech House, Boston, Mass., 1997), pp. 183–213.

M. Noguchi, T. Ishikawa, M. Ohno, S. Tachihara, “Measurements of 2-D birefringence distribution,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation, J. Tsujiuchi, ed., Proc. SPIE1720, 367–378 (1992).
[CrossRef]

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).
[CrossRef]

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurements Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 94–140.

J. Jaronski, “Theoretical and experimental study on corneal birefringence,” Ph.D. dissertation (Institute of Physics, Wrocław University of Technology, Wrocław, Poland, 1997).

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Figures (5)

Fig. 1
Fig. 1

Arrangement of the optical setup for the general configuration for phase-stepping imaging polarimetry.

Fig. 2
Fig. 2

Example of the optical setup for a 3 × 2 algorithm.

Fig. 3
Fig. 3

Examples of photoelastic measurements. Distribution of (a) azimuth angle α(x, y) (isoclinics) and (b) phase retardation δ(x, y) (isochromatics).

Fig. 4
Fig. 4

Examples of birefringence measurements of a quartz wedge. Distribution of (a) azimuth angle α(x, y) and (b) phase retardation δ(x, y).

Fig. 5
Fig. 5

Examples of birefringence measurements of the human cornea in vitro. Distribution of (a) azimuth angle α(x, y) and (b) phase retardation δ(x, y).

Tables (1)

Tables Icon

Table 1 Values of Azimuth Angles of Quarter-Wave Plate Q, ϕi, and βj for the 3 × 2 Algorithm

Equations (57)

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RE0=100eiϕ2222=2222 eiϕ.
S=Teiδ cos2 α+sin2 αeiδ-1sin α cos αeiδ-1sin α cos αeiδ sin2 α+cos2 α.
A=cos2 βsin β cos βsin β cos βsin2 β,
E=ASRE0
E=Tcos2 βsin β cos βsin β cos βsin2 β×eiδ cos2 α+sin2 αeiδ-1sin α cos αeiδ-1sin α cos αeiδ sin2 α+cos2 α×2222 eiϕ.
I=½ T2(1+sin 2α cos 2α1-cos δcos 2β+sin2 2α+cos2 2α cos δsin 2βcos ϕ+-sin 2α sin δcos 2β+cos 2α sin δsin 2βsin ϕ).
I=A+B cos ϕ+C sin ϕ,
A=½ T2,
B=½ T2sin 2α cos 2α1-cos δcos 2β+sin2 2α+cos2 2α cos δsin 2β,
C=½ T2cos 2α sin δ sin 2β-sin 2α sin δ cos 2β.
Iijx, y=Ajx, y+Bjx, yf1ϕi+Cjx, yf2ϕi,
Ajx, y=½ T2x, y,
Bjx, y=b1x, yf3βj+b2x, yf4βj,
Cjx, y=c1x, yf3βj+c2x, yf4βj,
b1x, y=½ T2x, ysin 2αx, ycos 2αx, y×1-cos δx, y,
b2x, y=½ T2x, ysin2 2αx, y+cos2 2αx, ycos δx, y,
c1x, y=-½ T2x, ysin 2αx, ysin δx, y,
c2x, y=½ T2x, ycos 2αx, ysin δx, y,
F1,2ϕiAjx, yBjx, yCjx, y=Gjx, y, ϕi.
F1,2ϕi=Ni=1N f1ϕii=1N f2ϕii=1N f1ϕii=1N f12ϕii=1N f1ϕif2ϕii=1N f2ϕii=1N f1ϕif2ϕii=1N f22ϕi,
Gjx, y, ϕi=i=1N Iijx, yi=1N Iijx, yf1ϕii=1N Iijx, yf2ϕi.
Ajx, yBjx, yCjx, y=F1,2-1ϕiGjx, y, ϕi.
F3,4βjb1x, yb2x, y=H1x, y, βj,
F3,4βj=j=1M f32βjj=1M f3βjf4βjj=1M f3βjf4βjj=1M f42βj,
H1x, y, βj=j=1M Bjx, yf3βjj=1M Bjx, yf4βj.
b1x, yb2x, y=F3,4-1βjH1x, y, βj.
c1x, yc2x, y=F3,4-1βjH2x, y, βj,
H2x, y, βj=j=1M Cjx, yf3βij=1M Cjx, yf4βi.
tan 2αx, y=-c1x, yc2x, y,
sin δx, y=2T2x, yc2x, ycos 2αx, y-c1x, ysin 2αx, y,
cos δx, y=2T2x, yc1x, yb1x, yc2x, y+b2x, y.
αx, y=12arctan-c1x, yc2x, y,
δx, y=arcsin2T2x, yc2x, ycos 2αx, y-c1x, ysin 2αx, y,
δx, y=arctanc2x, ycos 2αx, y-c1x, ysin 2αx, yc1x, yb1x, y/c2x, y+b2x, y,
Tx, y=2 j=1M Ajx, yM½.
ϕi=i2πN,  i=1N,
βj=jπM,  j=1M.
αx, y=12arctanI12x, y-I32x, yI11x, y-I31x, y,
δx, y=arcsinI11x, y-I31x, ycos 2αx, y+I12x, y-I32x, ysin 2αx, yTx, y
δx, y=arctank/l,
k=I12x, y-I32x, ysin 2αx, y+I11x, y-I31x, ycos 2αx, y,
l=I12x, y-I32x, yI12x, y-2I22x, y+I32x, y/I11x, y-I31x, y+I11x, y-2I21x, y+I31x, y,
Tx, y=½ I11x, y+I31x, y+I12x, y+I32x, y.
F1,2=3-10-110002,
Gjx, y=I1jx, y+I2jx, y+I3jx, y-I2jx, yI1jx, y-I3jx, y,
F1,2-1=0.50.500.51.50000.5.
Ajx, yBjx, yCjx, y=12I1jx, y+I3jx, yI1jx, y-2I2jx, y+I3jx, yI1jx, y-I3jx, y.
F3,4=1001,
H1x, y=-B2x, yB1x, y,
H2x, y=-C2x, yC1x, y.
F3,4-1=1001.
b1x, yb2x, y=-B2x, yB1x, y,
c1x, yc2x, y=-C2x, yC1x, y.
b1x, y=-½I12x, y-2I22x, y+I32x, y,
b2x, y=½I11x, y-2I21x, y+I31x, y,
c1x, y=-½I12x, y-I32x, y,
c2x, y=½I11x, y-I31x, y.

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